Lesson 5: Flux, etc.

1. Lesson 5: Flux, etc. • Flux determination (review) • Cell • Surface • Flux integral tallies (reaction rates) • K-effective calculations • Basis of PDF-modifying variance reduction

2. Cell flux estimation • Basic question: Why do we want to know the group flux in a cell? • Only reason: So that we can later turn it into some measurable (reaction rate, power distribution, dose) • Monte Carlo (rather perversely) is rather better at getting the reactions rates THEMSELVES • Two ways to get it: • After Monte Carlo gives you an incremental contribution to a reaction rate, back out the incremental flux that would have cause it and add it to a running total • Use an alternative flux definition to get flux directly

3. Cell flux estimation (2) • The first way to score flux is to add an incremental contribution every time there IS a collision in cell in by energy group (g): • then a collision contributes an “incremental” RR addition of 1 and an incremental flux addition of: • This is referred to as a “collision estimator”

4. Cell flux estimation (3) • Variation on this them is to score on particular TYPES of reactions and then score an amount depending on that REACTION’s cross section • Most common is an ABSORPTION estimator, which on each absorption event scores: • Another way to score flux is to go back to the basic definition of total macroscopic cross section:

5. Cell flux estimation (4) • Substituting this into the reaction rate equation gives us: • This is a “track length estimator” • Notice that the number of reactions has CANCELLED. • This estimator not only does NOT depend on an actual reaction occurring, but can even be used in a VACUUM

6. Cell flux estimation (5) • When to use which? General rules of thumb: • Track length estimator in thin regions • Collision estimator in high collision regions (especially scattering) regions • Absorption estimator in high absorption regions • Examples. Which estimator is most efficient for a: • Thin foils • Thick control rod (and thermal neutrons) • Diffusive low-absorber (e.g., D2O, graphite)

7. Surface flux estimation • A surface flux estimation (useful when you want to know the MAXIMUM dose in a room with an obvious highest-dose surface) is just a degenerate case of a track length estimator, for a cell with epsilon thickness:

8. Surface flux estimation • The equation breaks down to:

9. Reaction rate estimation • At first glance, this seems like a ridiculous question: • Reactions are basic events in a Monte Carlo simulation. • So, can’t you just COUNT them as they occur • Answer: Yes you can. But you might not want to. • The basic equation is, of course:

10. Reaction rate estimation (2) • Substituting our previous relation for flux, we get either: • Or

11. K-Effective Calculations • Fission in a subcritical situation—with a source—can (theoretically) can be handled as a multiple-particle-producing scattering reaction • The calculation of k-effective, however, is handled in a special way in Monte Carlo codes

12. K-Effective Calculations (2) • Source-less k-effective problems are solved by treating the particles coming out of fission as an external source • Problem: We know the particles’ energy and directional distributions but NOT their spatial distribution. • Solution: Instead of ONE problem, the calculation is handled as a SERIES of Monte Carlo problems, each of which uses the PREVIOUS problem’s fission sites as an external source of neutrons (and photons, if desired) • There are two difficulties: • How do we start the FIRST calculation • How do we deal with the fact that the fission spatial distribution is going to be TERRIBLE for a few rounds

13. K-Effective Calculations (3) • Procedure: • Make an initial guess of SPATIAL distribution of fission (Why not energy and angle?) • Use this guess as a source in a typical MC calculation (tallying new fission neutron production). • Estimate the eigenvalue the old fashioned way (fission neutron production in present cycle/previous cycle) • Use solution’s fission locations as next cycle’s source spatial information

14. K-effective Calculations (4) • Effects on calculation flow: • Problem is subdivided into (user-specified) number of cycles with a given number of source histories per cycle. • Problem delivers one eigenvalue guess per cycle. • As a practical matter, one discards the first few eigenvalue guesses until the fission spatial distribution “settles down” • Theoretically not satisfying since the cycles are obviously not independent. • Their dependence is smaller the larger the number of histories sampled in each cycle

15. Sample problem

16. Sample problem

17. Homework from text